,

Space-time resolved experiments for water waves

M. Chekroun, A. MaurelLOA/ Institut Langevin, ESPCI & CNRS, 10 rue Vauquelin, 75005 Paris, France

G. Lagubeau, V. PagneuxLaboratoire d’Acoustique de l’Universite du Maine & CNRS,

Avenue Olivier Messiaen, 72085 Le Mans, France

P. Cobelli, A. Przadka, P. PetitjeansPhysique et Mecanique des Milieux Heterogenes, ESPCI & CNRS, 10 rue Vauquelin, 75005 Paris, France

An overview of recent works on water wave propagation using a full time-space resolved methodis given. The experimental method allows us to precisely measure the surface elevation field withspatial and temporal resolutions given by the pixel size and frequency acquisition of an high speedcamera. Two typical problems are regarded: i) the propagation of water waves through surfacepiercing obstacles with trapped modes or directional emission, a problem of interest notably for itspractical applications to the protection of floating structures and to the canalization of the waterwave energy, ii) a study of water wave turbulence is also reported, exhibiting the interest to measurethe joint space-time power spectrum to study which hypothesis of weak turbulence theory survivesin a laboratory experiments.

I. INTRODUCTION

The very rich physics of waves is often difficult to measure due to the complexity in space and time of the phe-nomena encountered. Among the different domains (Electromagnetism, Quantum Mechanics, Acoustics, ...) whereexperiments on waves are carried out, the ability to measure the dynamics in space and time is rare because of veryshort time scale or small space scale. One kind of waves seems to be a good candidate to succeed in space-timeresolved measurements: these are gravity-capillary waves at the surface of water. Indeed, water waves present a directanalogy with linear scalar waves in other contexts of physics. Assuming the flow is irrotational and incompressible,the velocity potential φ(x, y, z, t) satisfies the Laplace equation ∆φ = 0, with appropriate boundary conditions. Onthe bottom z = −H0, the Neumann boundary condition to account for non penetrable bottom

∂zφ(x, y,−H0) = 0. (1)

The classical kinematic condition accounts for the condition that the vertical velocity must match the motion of thefree surface and dynamic boundary conditions on the free surface (basically, the Bernoulli equation expressed at thefree surface). Linearizing for small deformation of the surface, it yields in the frequency regime with e−iωt dependance,

∂zφ(x, y, 0) = −iωh(x, y),iωφ(x, y, 0) = gh(x, y),

(2)

with g the gravity constant and H(x, y) the local free surface height and H = H0 + h, where H0 is the height of thewater at rest. The explicit dependance in the z direction is

φ(x, y, z) = cosh k(z +H0) ϕ(x, y), (3)

with k satisfying the dispersion relation (for gravity waves)

ω2 = gk tanh kH0. (4)

The new two dimensional potential ϕ(x, y) is simply proportional to the wave field h(x, y) and the Laplace equationbecomes the 2D Helmholtz equation

(∆+ k2)h(x, y) = 0. (5)

When capillary effects are taken into account, it is sufficient to replace gk in equation (4) by (gk + γk3/ρ), where γis the surface tension constant.

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Besides, beyond the ability of water waves to experience similar behaviors to waves in other domains of physics,gravity-capillary waves have their own interest. The non-linearities of water waves produce spectacular phenomena.Tsunami or freak waves are examples that have received unceasing attention [1]. Also, the low wave speeds of waterwaves make easy the appearance of trans/supersonic wave regimes. For this reason, water waves have been used asthe analog of wave experiencing strong interactions, notably in quantum mechanics. As examples, the water wavespropagating on a moving counter flow is studied as the analog of quantum fields propagating around black holes [2];also, water waves interacting with a bathtub vortex is studied as the analog with a beam of particles interacting witha magnetic field, a problem known in quantum mechanics as the Aharonov-Bohm effects [3].

Recently, we proposed the application of the Fourier Transform Profilometry (FTP) [4] to study different cases ofcomplex wave phenomena with water waves [5–9]. The FTP is a technique based on the analysis of the deformation offringes projected onto the surface whose shape measurement is wanted. This yields a measurement of the deformationof the surface, in our case the wave field. Coupled to a high speed camera, it is able to produce a ”movie” of themeasured wave field with a very good resolution both in space and time.

We report in this paper experiments of gravity waves in different contexts. In the first part, we focus on the problemof linear water wave propagating through obstacles that are surface piercing scatterers. This configuration is used toshow the existence of trapped modes localized in the vicinity of the scatterers and also to test the ability to obtaindirectional emission with a cluster of periodically distributed scatterers. In the second part, we present a statisticalanalysis of non-linear random waves. There, the space-time resolved measurements obtained owing to our techniqueallows us to study the joint space-time power spectrum.

II. EXPERIMENTAL SET-UP, FTP MEASUREMENTS

The experimental set up consists in a tank filled with water in which different contexts of wave propagation areproduced (Fig. 1). The main originality of our set up is the full space-time resolved method of measurement of thesurface elevation. Beyond the usual visualizations or measurements at one point, we developed an optical method ofmeasurement of the free surface. The method has been described in details in the references [4–6]. It is based onthe analysis of the deformation of fringes projected on the free surface of a light diffusive liquid. The surface depthinformation is encoded into a deformed fringe pattern recorded by the acquisition sensor, allowing it to be measuredby comparison to the original (undeformed) grating image. It is therefore the phase shift between the reference anddeformed images which contains all the information of the deformed surface.

We proposed recently significant improvements of the technique 1) by developing a filter free demodulation method[10] and 2) by analyzing carefully the properties of the diffusive liquid to find suspensions that allow to make highdiffusivity with, when needed, the same properties as clean water, namely weak attenuation of the gravity-capillarywaves [11]. As it is now, the method permits us to measure the 2D field with: 1) the spatial and temporal resolutionsof the camera (in our case, a v9 phantom high speed camera with up to more than 1000 fps) and 2) with an accuracyin the measured height given by the periodicity of the projected fringes (in our experiments, up to 10 µm accuracyhas been obtained [9]).

III. PROPAGATION OF WATER WAVES THROUGH OBSTACLES

The problem of the propagation of water waves through an ensemble of scatterers has many applications in navaland coastal engineering. Among these, water wave resonances, as can be observed near floating structures hasreceived much attention due to its practical application. Also of interest is the possibility to capture the energy ofthe ocean waves. This problem has been recently revisited in the framework of the design of metamaterials comingfrom the community of electromagnetism within the general question of wether or not we are able to control and tocanalize water waves. We present in this section experimental studies of these two problems owing to our quantitativemeasurements of the water wave fields. The aim in studying experimentally such configurations is to test how robustare the theoretical or numerical predictions, based on the 2D Helmholtz equation with idealized boundary condition (asNeumann boundary conditions on the wall) and idealized flow (irrotational, incompressible). As will be seen, the caseof a unique surface piercing cylinder (Section A) appears to nicely follow these approximations. To the opposite, thecollective effects appearing when several cylinders are considered may considerably affects the theoretical predictions(Section B). Of course, the conclusions that we can make regarding the result of a laboratory experiments are notnecessary applicable at the ocean scale. This question is addressed in Section B.

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FIG. 1: Experimental set-up. The main tank is filled with a light diffusive liquid. The method for free surface measurementneeds a projector for the fringe projection and an acquisition sensor (camera) as on the top of the photography.

A. Wave resonance near an obstacle

In the framework of the classical theory of linearized water waves in unbounded domains, trapped modes consistin non-propagative localized oscillation modes of finite energy occurring at some well-defined frequency and which, inthe absence of dissipation, persist in time even in the absence of external forcing. The first theoretical example of suchtrapping mode in the theory of water waves, due to Stokes [12], corresponds to waves which travel in the long-shoredirection over a uniformly sloping beach and decay to zero in the seaward direction. A recent review on edge wavescan be found in [13] and a description of edge waves in an oceanographic context in [14].

The fluid around a free surface piercing circular cylinder in a long narrow wave tank can exhibit a local oscillation.In practice, the occurrence of near-trapped modes is of particular relevance to offshore structures based on a largenumber of piles (such as oil rigs, very large floating structures and breakwaters) as it implies the presence of verylarge loads on individual elements of the array. Recent progress in the understanding of wave interaction with arraysof offshore structures, resonant effects and their consequences for finite arrays has been discussed by [15].

FIG. 2: Left figure: Experimental set-up. Symmetric and antisymmetric modes are generated at the entrance of the waveguideby the wavemaker. A free surface piercing circular cylinder of diameter 2a ∈ [2, 10] cm lies in the center of the waveguide ofwidth 2d = 10 cm. Right Fig.: Typical instantaneous field of the surface elevation hT (x, y, t), here for a/d = 0.50, kd = 1.32.The scale of the colorbar is in mm.

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We characterized the trapped mode and we report here the most relevant conclusion of our work (for details see[7, 8]). Our experimental set-up consists of a water tank with constant water level at rest which is chosen to be fixedat H0 = 5 cm (Fig. 2). The system of interest is placed inside the tank: a waveguide formed by two parallel verticalwalls 60 cm long, a distance 2d = 10 cm apart, has a free surface piercing vertical circular cylinder of diameter 2a (a =1 to 5 cm) located symmetrically between the two walls. Water waves are generated by a wavemaker forming an angleof around 45o with the waveguide axis Ox (so symmetric and antisymmetric modes are generated at the entranceof the waveguide) at a frequency f ∈ [2, 3] Hz. Typical free surface deformation fields hT (x, y, t) near resonance areshown in fig. 2. The acquisition rate of the camera is synchronized with the wavemaker in order to get 200 acquisitionshT (x, y, t) over two periods of the water wave oscillation. For a perfect fluid, in absence of dissipation, the dispersionrelation for the water waves is given by ω = gk tanh kH0 where ω is the driving pulsation, k the wave number andg = 9.81 m.s−2 (the effects of surface tension are neglected).

In order to obtain quantitative characteristics of the resonance, we proceed as follow: the linear part of the signalis obtain by extracting at each position (x, y), the coefficient h1(x, y) of the Fourier series

hT (x, y, t) =�

n

Re�hn(x, y)e

inωt�, (6)

(Re denotes the real part) and we find that the weight of the non-linearities |hT − h1|/|hT | is less than 15 %. h1 isthen separated into an even part he and an odd part ho. The even field he makes the propagative plane mode toappear, that can be used to get a direct measurement of the wave number k. The agreement with the theoreticaldispersion relation (not reported here) is of about 2%.

FIG. 3: Left: reduction of the problem to a 1D problem. The modal component h1(x) of the first antisymmetric mode(evanescent, k1 = iα) in the far field allows to get the reflexion R and transmission T coefficients, Eq. (8), (bottom curve isa typical example of the experimental h1(x) behavior in a log-lin representation. Right: Example of the resulting resonancecurve as a function of the frequency for a/d = 0.9, two resonances kc

The odd part ho is then analyzed to quantify the behavior of the trapped mode. If correctly governed by theHelmholtz equation (∆+k2)h1(x, y) = 0, with Neumann boundary condition at the walls, the field can be decomposedonto the modes in the transverse direction sinnπy/2d (n integer). The odd part of the field is composed of evanescentmodes only, so that we expect only the resonant mode to be dominant in the far field of the cylinder and the problemreduces to a 1D problem (In the near field, the higher transverse modes are expected to contribute to the 2D solution)

ho(x, y) � a1(x) sinπy/2d, (7)

and for this evanescent mode

a1(x < 0) = Ae−αx +AReαx,a1(x > 0) = ATe−αx,

(8)

with A the amplitude of the incident wave, α =�

k2 − π2/(2d)2 the wavenumber of the first evanescent mode and(R, T ) the reflection and transmission coefficients (Fig. 3). The reflection and transmission coefficients (R, T ) arefitted for each frequency, leading to the resonance curves, as exemplified in fig. 3 for a/d = 0.9 (two resonances arevisible). As is obvious from these curves, the classical Breit-Wigner resonance shape is not pertinent since it is not

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FIG. 4: Resonance frequency kcd as a function of the aspect ratio a/d. Open symbols are our experimental measurements,plain lines are the theoretical predictions by [17, 18].

able to reproduce the asymmetry, a behavior already observed for resonances occurring near the cutoff frequency in[16] and that can be described using a particular resonance shape of the form T = B/(1− C/(αd)). Typical value ofthe attenuation part of the wavenumber used to fit our resonance curves is 1.5 m−1. This value is in good agreementwith careful measurements of the attenuation in our experiments [11].

The behavior of the resonance frequency kcd when changing the size of the cylinder is shown in Fig. 4, thatsummarizes our main result: we observe two branches. The first corresponds to trapped modes symmetric withrespect to Oy-axis and the second to trapped modes antisymmetric with respect to Oy-axis. The experimental resultsare compared to the theoretical predictions of ref. [17] for the first branch and with the theoretical predictions of ref.[18] for the second branch. This second branch is predicted to exist for a/d > 0.81 and it is experimentally confirmedhere. Globally, an excellent agreement is observed with the theoretical predictions for both branches. It is worthnoting that this is the case even for the largest values of a/d, for which the effect of a meniscus in the small regionbetween the cylinder and the waveguide walls might be important. As will be seen in the next section, the dynamicsof the contact line at the boundaries are not always negligible.

B. On the possibility of enhanced and directional water wave emission by a periodic lattice of cylinders.

Photonic and phononic crystals have attracted considerable attention due to the existence of frequency band gaps,and the particular behavior of waves that are propagating in corresponding periodical structures. For applicationto water waves, interesting phenomena have been reported such as the superlensing and self-collimation phenomena[19, 20]. In a recent paper, Mei and co authors [21] proposed the design of a highly directional liquid surface wavesource by utilizing the high density of states at the band-edge frequency of a periodic structure. Their numericalstudy exhibits a high directivity associated to a resonance in the wave intensity transmitted by the lattice in onedirection. We report below the experimental study of the proposed configuration. It appears that this effect is notrobust in our laboratory scale experiments. The measured patterns of transmitted wave intensity nicely compares withnumerical calculations but the agreement needs an unexpected high attenuation to be accounted for in the numericalmodel. This can be due to the attenuation produced by the dynamics associated with moving contact lines [22], aphenomenon that can be less important at the ocean scale.

The experimental set-up is shown on Fig. 5. The lattice used in the experiment is composed of brass cylinders ofdiameter 2a = 2 cm forming a square lattice with lattice step d = 3 cm (filling fraction πa2/d2 = 0.35 as consideredin [21]). The cylinders are placed (immerged) in a wave tank with a water depth H0 = 3.3 cm. The point source isproduced by a thin tip controlled by a linear motor. The tip oscillates vertically at the free surface with a sinusoidalfrequency ω. Results presented below corresponds to ω = 38.51 s−1, near the expected resonance at 35 s−1. Similar

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FIG. 5: Experimental set-up: a square lattice of 6×6 cylinders are placed in a wave tank. The point source is produced by athin tip oscillating vertically at a sinusoidal frequency ω.

FIG. 6: Left: Direct visualization of the wave field, Center: instantaneous measured field and Right: Linear part of themeasured field (colorbar on (b) and (c) are in mm).

results have been observed at other frequencies [23].Example of measured instantaneous field hT (x, y, t) using our FTP method is shown on Fig. 6(b). After filtering the

non-linearities, the complex field h1(x, y) is obtained (Fig. 6(c), see Eq. (6)). This allows us to get the wave intensitypattern |h1|2, as illustrated in Fig. 7. The measured water wave intensity can be compared to the pattern calculatednumerically when considering h1 satisfies the Helmholtz equation with Neumann boundary condition on the cylinderboundaries (this is done by using multiple scattering simulations as in [21]). We performed a simulation in which theunique adjustable parameter is the attenuation of the water wave encapsulated in the wavenumber, assumed to havean imaginary part αnum. For αnum = 10 m−1, an excellent agreement is found, both on the 2D pattern and on thescattering function (wave amplitude h1(r, θ) � f(θ)/

√r measured at the same distance r from the source with θ the

angular direction). This is illustrated on Fig. 7. As previously said, the attenuation in our experiments has beencarefully characterized [11] and we found α � 3 m−1 for ω = 38.51 s−1, three times smaller than the one that we haveto take in the numerics to reproduce the experimental measurements. The enhancement of the apparent attenuationproduced by the dynamics associated with moving contact lines has been studied in details, see [22] and referencetherein. However, in the experiments with a unique cylinder in a waveguide, presented in the previous section, noenhancement of the attenuation has been found. This suggests that the vicinity of the cylinders produces a moreimportant effects of the multiple moving contact lines and this leads to the increasing of the attenuation. However,

FIG. 7: Patterns of water wave intensity as observed Left: experimentally and Center: in the numerics considering an attenu-ation αnum = 10 m−2. Right: Comparison of the scattering functions experimentally measured and obtained numerically.

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FIG. 8: Directivity expected from numerical calculations (MST, as used in [21]) without attenuation αnum =0. Left: intensitypattern and Right: scattering function for ω = 38.51 s−1. The position of the source is indicated by a white star.

the agreement found between numerics and experiments shows that this phenomenon can be simply encapsulated inan extra attenuation, the behavior of the water wave being otherwise very well described by the Helmholtz equation,as considered in numerical calculations.

Finally, Fig. 8 shows the wave intensity pattern and the scattering function at the same frequency but neglectingthe attenuation (αnum= 0). The directivity that is observed in that case corresponds to the one described in [21]. Itappears clearly by comparison with Fig. 7 that the directivity is lost because of the attenuation.

IV. WAVE TURBULENCE

Wave turbulence (WT) theory predicts that non-linear random waves are able to experience an energy cascadethrough different scales, similar to the famous Kolmogorov cascade in classical hydrodynamics. In the context ofgravity wave [24, 25], the power spectrum is expected to follows

|ηω|2 ∝ P 1/3ω−4, |ηk|2 ∝ P 1/3k−5/2, (9)

in the inertial range (P is the injected energy) and this law seems to reasonably fit the spectra measured for oceanwaves in many cases [26, 27]. To get deeper insights into the non-linear behavior of gravity waves, well controlledlaboratory experiments have been developed in the recent years [28–31]. However, they have been unable to reproducethe spectrum at the oceanic scale, and a question has emerged on wether or not the conditions for the applicability ofWT theory can be in place in such finite systems. Indeed, this theory assumes weak non-linearities, low attenuationand small finite size effects and meeting these requirements is particularly difficult in a laboratory tank [32, 33].

We report here results in a laboratory experiment showing that it is possible to recover the spectrum as predictedwithin WT theory. This is done using a low frequency forcing bandwidth (as narrow as possible). Indeed, assuggested in [33], this should enlarge the inertial range in the gravity regime. In the presented experiments, thefrequency bandwidth has the maximum frequency ωm = 9.44 s−1, for a transition between gravity and capillarywaves at ωc = 85 s−1 (kc ≡

�ρg/γ = 369 m−1 ). The water waves are generated by two wavemakers in a (177 × 61)

cm2 tank filled with water with depth at rest H0 = 5 cm. The wavemaker motions are controlled by a random signalwithin a broadband frequency range [0, 9.44] s−1 with maximum amplitude A = 30 mm. Also, in these experiments,the sensitivity of the FTP measurement is improved by treating the phase shifts between two successive images (ratherthan the phase shift with a reference to unperturbed free surface), resulting in the measurement of the velocity of thesurface elevation η(r, t). A typical velocity measured field is shown in Fig. 9.

Our space-time measurement allows to experimentally measure the joint space-time power spectrum

|ηk,ω|2 ≡�

dkdω �η(r�, t�)η(r+ r�, t+ t�)� ei(ωt+k.r), (10)

a quantity that shows how the non-linear interactions spread the wave energy in the 3D (k,ω)- space. This allows us totest the validity of the premise of WT theory that assumes the concentration of the wave energy on the renormalizeddispersion relation (which remains close to the linear dispersion relation) and it allows us to evaluate the isotropyof the energy distribution. This is illustrated on Fig. 10. Different features can be seen: on the one hand, theenergy is indeed spreads isotropically whatever being the considered frequency, which indicates a weak dependanceon the tank shape. On the other hand, the energy remains concentrated on the dispersion relation manifold. Theseare indications that the conditions of applicability of WT theory are satisfied in our experiments. Indeed, it hasbeen observed numerically [33] and experimentally [34, 35] that non-linear mode, also called bound waves, can exist.

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FIG. 9: Typical instantaneous velocity field η(r, t) (color scale are in m/s) for A= 28 mm).

FIG. 10: Left: |ηk,ω|2 for (a) ω = 24 s−1, (b) ω = 36.4 s−1, (c) ω = 49 s−1 and (d) ω = 62 s−1 (A = 3 mm), illustrating theisotropy in the (kx, ky)-space. Right: Typical spectrum |ηk,ω|2 at forcing amplitude A = 20 mm. The inset shows the collecteddispersion relations for A = 1, 6, 12, 20 and 28 mm. The dashed line shows the linear dispersion relation and the dotted lineshows the the dispersion relation of the first bound waves.

These modes correspond to strong non-linearities with frequencies nω. They appear near the injection scale and theirexistence may affect the behavior of the power spectrum.

Finally, the most significant feature is shown on Fig. 11: a forcing independent inertial range is observed with ascaling law in ω in agreement with the ZF predictions |ηω|2 ∝ ω−4. The spectra in the k-space are shown in the inset,where the ZF prediction |ηk|2 ∝ k−3/2 are displayed for comparison.

Other experiments (not reported here) have been performed using an higher ωm = 25.13 s−1. In that case, boundwaves are observed and this is accompanied with the lost of the forcing independent inertial range. Instead, in theregime of gravity wave, the power spectrum is found to have an increasing spectral slope, from -5 to -4 when increasingthe forcing amplitude. This behavior is the one usually observed in laboratory experiments [30–32].

V. CONCLUDING REMARKS

We have shown that space-time resolved measurements of complex wave phenomena can be obtained by usingFourier Transform Profilometry for water waves.

When the interest is in the characterization of linear waves in the frequency regime, our method permits to filter thesignal at the driving frequency. We exemplified this by the study of the wave scattering by Neumann type scatterers:We characterized precisely the trapped modes around a surface piercing cylinder in a waveguide. Also, we have shownthat the directional emission theoretically displayed for a source in a cluster of periodically arranged cylinders is notrobust with respect to the attenuation of water in mesoscale laboratory experiments.

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FIG. 11: Spectra |ηω|2 for several forcing amplitude (between A= 1 to 30 mm). Vertical dotted lines at 30 s−1 and at ωc=85s−1 delimitate the inertial range of the gravity regime.Top inset: corresponding spectra |ηk|2. Bottom inset: spectral exponentas a function of A (multiple symbols correspond to multiple fits in a moving window within the inertial range).

The study of wave turbulence corresponds to a case where the nonlinear interactions govern the dynamics of thewave field. In that case, the full space-time characterization is of particular importance. For wave turbulence, itpermits to have access to the (k,ω)-space that is the natural space in turbulence theory. We showed that our methodis very efficient: the resolution was sufficient to calculate the experimental joint space-time spectrum. A statisticalanalysis of this spectrum showed a good agreement with the cascade scenario of wave turbulence theory.

This work is supported by the ANR Tourbillonde.

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